"But circles are 2D", I hear you cry, "aren't we dealing with 3D?". If the circumference is less than expected, the space is positively curved, if it is greater, space is negatively curved. We can measure how 3D space is curved by using the circle example above. What about in three-dimensional space? This is harder to comprehend because the human mind has trouble imagining things outside of three dimensions. If we draw a circle around the North Pole at a distance of x, the circumference will be less than expected since the radius of the circle is actually the perpendicular distance between the circle and the central axis of the Earth, not x. When it is less than zero, space is negatively curved.Īnother way to measure the curvature of a space is to measure the circumference of a circle and see how it compares with the circumference predicted by the formula 2 PI * radius. When this excess angle is greater than zero, the space is said to postively curved. You have drawn a polygon with three straight sides, hence it is a triangle, but the internal angles add up to 270°. This will bring you back to your starting point. Turn back towards the North Pole and draw another line of length x. Turn 90° and draw a line of length x along the equator. To give an example, imagine starting at the North Pole and drawing a straight line to the Equator, which has length x. In fact, the excess angle is proportional to the area of the triangle.
![3 dimensional curved space 3 dimensional curved space](https://static.artfido.com/2018/08/line-drawing-artist-002.jpg)
![3 dimensional curved space 3 dimensional curved space](https://image.slidesharecdn.com/threedimensionalspace-dfs-130119175330-phpapp02/95/three-dimensional-space-dfs-17-638.jpg)
If you draw a polygon with three straight sides, the internal angles will sum to greater than 180°.
![3 dimensional curved space 3 dimensional curved space](https://cdn1.byjus.com/wp-content/uploads/2021/07/coordinate-planes-i-n-three-dimensional-space.png)
The internal angles of a triangle drawn on that plane will always sum to 180°. It is easier to explain this using two-dimensional examples because we can look into it from the outside. For example, the angles of a triangle add up to more than 180°. Curved space is defined to be anywhere where the rules of geometry 'do not work'.